Heat oil in a nonstick pan or wok and add in shallots. Cook over high heat, stirring frequently, until shallots are completely soft or about 12 minutes. Meanwhile, line a rimmed baking sheet with 6 layers of paper towels. Continue to cook, stirring constantly, until shallots are light golden brown, about 8 minutes. Strain the hot oil into a heatproof bowl or saucepan. Set shallot oil aside and transfer shallots onto paper towels. Blot to absorb excess oil. Season shallots well with salt. Allow to cool completely, about 45 minutes. Once they have cooled, transfer shallots to a sealed airtight container.
Smash mushrooms under bottom of large skillet until broken into 1/4- to 1/2-inch pieces. Roughly chop into pieces approximately 1/8 to 1/4 inch in size. Set aside. Combine soy sauce, lemon juice, chicken stock, and heavy cream in a 1-quart liquid measure or medium bowl. Add 2 tablespoons reserved shallot oil (save remainder in an airtight container for another use) plus butter to a large nonstick skillet. Heat over high heat until butter is melted and foaming subsides. Add mushrooms to skillet and cook, stirring occasionally, until liquid is evaporated and mushrooms begin to sizzle, 6 to 10 minutes. Reduce heat to medium-high. Add garlic and cook, stirring constantly, until fragrant, about 30 seconds. Add flour and cook, stirring constantly, until light golden blond, 1 to 2 minutes. Whisking constantly, add stock/cream mixture. Bring to a boil, reduce to a simmer, and cook until mixture achieves a texture between that of pancake batter and heavy cream, about 5 minutes. Season to taste
Adjust oven rack to lower-middle position and preheat oven to 350°F. Bring 1 gallon water and 1/4 cup kosher salt to a boil over high heat. Fill a large bowl with 1 quart ice cubes and 2 quarts water. Add green beans to water and boil until tender but still bright green, about 5 minutes. Drain through a colander set over the sink and immediately transfer to ice water until completely cooled. Drain green beans and set aside.
Combine green beans, mushroom sauce, and 1 cup fried shallots. Transfer to a 9- by 13-inch rectangular baking dish or a 10- by 14-inch oval baking dish. Bake until hot and bubbly, 15 to 20 minutes. Top with remaining cup fried shallots and serve.